The quality of various Hawkes-process-based order book models are assessed using some objective criteria. We start with a precise empirical analysis of the dependencies between order arrivals of various types, then, models built from multivariate, possibly nonlinear, Hawkes processes with multiple exponential kernels are introduced. Models are evaluated based on the distribution of forward recurrence times and the signature plot. This approach allows us to discriminate between various Hawkes-process- based models, and provide a financial interpretation of the more successful ones in terms of their behaviour at various time scales, and the presence of inhibition as well as excitation effects.